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The question is asking which statement is correct for "Galerkin’s procedure."

Galerkin's method is a widely used technique in numerical methods, particularly for solving partial differential equations. In this method, the weighting functions (also called test functions) are chosen to be the same as the basis functions used for approximating the solution. This ensures that the residual is orthogonal to the space spanned by the basis functions.

Let’s analyze the options:

a. **Basis function and weighting function both are sinusoidal**: This is not necessarily true for Galerkin’s method. While sinusoidal functions could be chosen as basis and weighting functions in specific cases, it is not a defining characteristic of the method.

b. **Weighting function is sinusoidal**: Again, this is not a general requirement in Galerkin's method. The weighting functions are chosen to match the basis functions but do not need to be sinusoidal specifically.

c. **Weighting function is equal to delta function \((\omega_n = \delta(p - p_m))\)**: This suggests a method like collocation, where the weighting functions are delta functions that force the residual to be zero at specific points. This is not true for Galerkin's method.

d. **Weighting function is equal to basis function \((\omega_n = g_n)\)**: This is the correct statement. In Galerkin’s method, the weighting functions are chosen to be the same as the basis functions used in the approximation of the solution.

### Answer: d. Weighting function is equal to basis function \((\omega_n = g_n)\)

Would you like more details on this procedure or its applications? 

Here are five related questions for further exploration:
1. How does Galerkin’s method differ from the collocation method?
2. What are common choices for basis functions in Galerkin’s method?
3. How is Galerkin’s method applied to finite element analysis?
4. Can you explain the concept of residuals in the context of Galerkin’s method?
5. What are the advantages of using Galerkin's method in solving differential equations?

**Tip:** When applying Galerkin's method, it's important to choose basis functions that reflect the behavior of the problem you're trying to solve for better accuracy.

Which of the following is a correct statement for Galerkin's procedure?

Learn about Galerkin’s method, a technique used in numerical methods for solving partial differential equations. Understand the relationship between basis functions and weighting functions, and why they are the same in Galerkin’s procedure. Suitable for university-level engineering and mathematics students.

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